Hybrid Projection Method for Generalized Mixed Equilibrium Problems,Variational Inequality Problems and Fixed Point Problems in Banach Spaces

WANG Ya-qin; ZENG Lu-chuan

doi:10.3879/j.issn.1000-0887.2011.02.012

Volume 32 Issue 2

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2011 > 32(2): 241-252

WANG Ya-qin, ZENG Lu-chuan. Hybrid Projection Method for Generalized Mixed Equilibrium Problems,Variational Inequality Problems and Fixed Point Problems in Banach Spaces[J]. Applied Mathematics and Mechanics, 2011, 32(2): 241-252. doi: 10.3879/j.issn.1000-0887.2011.02.012

Citation:

PDF( 395 KB)

Hybrid Projection Method for Generalized Mixed Equilibrium Problems,Variational Inequality Problems and Fixed Point Problems in Banach Spaces

doi: 10.3879/j.issn.1000-0887.2011.02.012

WANG Ya-qin¹,
ZENG Lu-chuan²

Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, P. R. China;
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China

Received Date: 2010-10-25
Rev Recd Date: 2010-12-30
Publish Date: 2011-02-15

Abstract

Abstract

A new hybrid projection iterative scheme was introduced for approximating a common elementof the solution set of a generalized mixed equilibrium problem,the solution set of a variational inequalityproblem and the set of fixed points of a relatively weak nonexpansive mapping in Banach spaces. The results obtained generalize and improve the recent ones announced by many others.

FullText(HTML)

References(16)

References

[1]	Takahashi W.Nonlinear Functional Analysis[M]. Tokyo: Kindikagaku, 1988.(in Japanese).
[2]	张石生.Banach空间中广义混合平衡问题[J]. 应用数学和力学, 2009, 30(9):1033-1041.(ZHANG Shi-sheng.Generalized mixed equilibrium problem in Banach spaces[J].Applied Mathematics and Mechanics(English Edition), 2009, 30(9):1105-1112.)
[3]	CENG Lu-chuan, YAO Jen-chih.A hybrid iterative scheme for mixed equilibrium problems and fixed point problems[J]. J Comput Appl Math, 2008, 214(1): 186-201. doi: 10.1016/j.cam.2007.02.022
[4]	Takahashi S, Takahashi W.Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space[J].Nonlinear Anal, 2008, 69(3): 1025-1033. doi: 10.1016/j.na.2008.02.042
[5]	Habtu Zegeye, Naseer Shahzad.Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings[J]. Nonlinear Anal, 2009, 70(7):2707-2716. doi: 10.1016/j.na.2008.03.058
[6]	Matsushita S Y, Takahashi W.A strong convergence theorem for relatively nonexpansive mappings in a Banach space[J]. J Approxim Theory, 2005, 134(2): 257-266. doi: 10.1016/j.jat.2005.02.007
[7]	Takahashi W, Zembayashi K.Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces[J].Nonlinear Anal, 2009, 70(1): 45-57. doi: 10.1016/j.na.2007.11.031
[8]	CHANG Shi-sheng, Lee Joseph H W, CHAN Chi-kin.A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications[J].Nonlinear Anal, 2010, 73(7): 2260-2270. doi: 10.1016/j.na.2010.06.006
[9]	Alber Y I. Metric and generalized projection operators in Banach spaces: properties and applications[C]Kartsatos A G.Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. New York: Dekker, 1996: 15-50.
[10]	Kamimura S, Takahashi W.Strong convergence of proximal-type algorithm in a Banach space[J]. SIAM J Optim, 2002, 13(3): 938-945. doi: 10.1137/S105262340139611X
[11]	Reich S. A weak convergence theorem for the alternating method with Bergman distance[C]Kartsatos A G.Theory and Applications of Nonlinear Operators of Accretive and Monotone Type.New York: Dekker, 1996: 313-318.
[12]	Butanriu D, Reich S, Zaslavski A J.Asymtotic behavior of relatively nonexpansive operators in Banach spaces[J].J Appl Anal, 2001, 7(2): 151-174.
[13]	Butanriu D, Reich S, Zaslavski A J.Weakly convergence of orbits of nonlinear operators in reflexive Banach spaces[J].Numer Funct Anal Optim, 2003, 24(5): 489-508. doi: 10.1081/NFA-120023869
[14]	XU Hong-kun.Inequalities in Banach spaces with applications[J].Nonlinear Anal, 1991, 16(12):1127-1138. doi: 10.1016/0362-546X(91)90200-K
[15]	Pascali D, Sburlan S.Nonlinear Mappings of Monotone Type[M].Bucaresti, Romania: Editura Academiae, 1978.
[16]	Rockfellar R T.Monotone operators and the proximal point algorith[J].SIAM J Control Optim, 1976, 14(5): 877-898. doi: 10.1137/0314056

Relative Articles

[1]	WANG Xiaoting, LONG Xianjun, PENG Zaiyun. A Double Projection Algorithm for Solving Non-Monotone Variational Inequalities[J]. Applied Mathematics and Mechanics, 2022, 43(8): 927-934. doi: 10.21656/1000-0887.420414
[2]	YANG Jun. A Golden Ratio Algorithm for Solving Nonmonotone Variational Inequalities[J]. Applied Mathematics and Mechanics, 2021, 42(7): 764-770. doi: 10.21656/1000-0887.410359
[3]	SHAO Chongyang, PENG Zaiyun, WANG Jingjing, ZHOU Daqiong. Characterizations of HContinuity for Solution Mapping to Parametric Generalized Weak Vector QuasiEquilibrium Problems[J]. Applied Mathematics and Mechanics, 2019, 40(4): 452-462. doi: 10.21656/1000-0887.390198
[4]	ZHAO Shilian. Strong Convergence of CQ Algorithms for Split Feasibility Problems in the Hilbert Spaces[J]. Applied Mathematics and Mechanics, 2019, 40(1): 108-114. doi: 10.21656/1000-0887.390012
[5]	YANG Li, LI Jun. Modified Halpern Iteration and Viscosity Approximation Methods for Split Feasibility Problems in Hilbert Spaces[J]. Applied Mathematics and Mechanics, 2017, 38(9): 1072-1080. doi: 10.21656/1000-0887.380106
[6]	ZENG Jing, PENG Zai-yun, ZHANG Shi-sheng. Existence and Hadamard Well-Posedness of Solutions to Generalized Strong Vector Quasi-Equilibrium Problems[J]. Applied Mathematics and Mechanics, 2015, 36(6): 651-658. doi: 10.3879/j.issn.1000-0887.2015.06.009
[7]	TANG Guo-ji, HUANG Nan-jing. A Projected Subgradient Method for Non-Lipschitz Set-Valued Mixed Variational Inequalities[J]. Applied Mathematics and Mechanics, 2011, 32(10): 1254-1264. doi: 10.3879/j.issn.1000-0887.2011.10.011
[8]	DING Xie-ping. Bilevel Generalized Mixed Equilibrium Problems Involving Generalized Mixed Variational-Like Inequality Problems in Reflexive Banach Spaces[J]. Applied Mathematics and Mechanics, 2011, 32(11): 1361-1377. doi: 10.3879/j.issn.1000-0887.2011.11.010
[9]	ZHANG Shi-sheng, LEE Heung-wing Joseph, CHAN Chi-kin, LIU Jing-ai. Viscosity Approximation With Weak Contractions for Fixed Point Problem Equilibrium Problem and Variational Inequality Problem[J]. Applied Mathematics and Mechanics, 2010, 31(10): 1211-1219. doi: 10.3879/j.issn.1000-0887.2010.10.008
[10]	ZHANG Shi-sheng, RAO Ruo-feng, HUANG Jia-lin. Strong Convergence Theorem for a Generalized Equilibrium Problem and k-Strict Pseudocontraction in Hilbert Spaces[J]. Applied Mathematics and Mechanics, 2009, 30(6): 638-647. doi: 10.3879/j.issn.1000-0887.2009.06.002
[11]	LIU Ying. Strong Convergence Theorems for Relatively Nonexpansive Mappings and Inverse-Strongly-Monotone Mappings in a Banach Space[J]. Applied Mathematics and Mechanics, 2009, 30(7): 865-872. doi: 10.3879/j.issn.1000-0887.2009.07.011
[12]	ZHANG Shi-sheng. On the Generalized Mixed Equilibrium Problem in Banach Spaces[J]. Applied Mathematics and Mechanics, 2009, 30(9): 1033-1041. doi: 10.3879/j.issn.1000-0887.2009.09.004
[13]	ZHU Li, LI Xiu-hua, GOU Xing-ming. Optimal Obstacle Control Problem[J]. Applied Mathematics and Mechanics, 2008, 29(5): 505-514.
[14]	XU Hai-li, GUO Xing-ming. Auxiliary Principle and Three-Step Iterative Algorithms for Generalized Set-Valued Strongly Nonlinear Mixed Variational-Like Inequalities[J]. Applied Mathematics and Mechanics, 2007, 28(6): 643-650.
[15]	DING Xie-ping, LIN Yen-cherng, YAO Jen-chih. Three-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities[J]. Applied Mathematics and Mechanics, 2007, 28(8): 921-928.
[16]	Ding Fangyun, Zhang Xin, Ding Rui. Boundary Mixed Variational Inequality in Friction Problem[J]. Applied Mathematics and Mechanics, 1999, 20(2): 201-210.
[17]	Gao Yang. Contact Problems and Dual Variational Inequality of 2-D Elastoplastic Beam Theory[J]. Applied Mathematics and Mechanics, 1996, 17(10): 895-908.
[18]	Zheng Tie-sheng, Li Li, Xu Qing-yu. An Iterative Method for the Discrete Problems of a Class of Elliptical Variational Inequalities[J]. Applied Mathematics and Mechanics, 1995, 16(4): 329-335.
[19]	Zeng Lu-chuan. Iterative Algorithms for Finding Approximate Solutions of Completely Generalized Strongly Nonlinear Quasivariational Inequalities[J]. Applied Mathematics and Mechanics, 1994, 15(11): 1013-1024.
[20]	Zhang Cong-jun. Generalized Variational Inequalities and Generalized Quasi-Variational Inequalities[J]. Applied Mathematics and Mechanics, 1993, 14(4): 315-325.